The Congruent Number Problem

THE CONGRUENT NUMBER PROBLEM

KEITH CONRAD

1. Introduction A right triangle is called rational when its legs and hypotenuse are all rational numbers. Examples of rational right triangles include Pythagorean triples like (3, 4, 5). We can scale such triples to get other rational right triangles, like (3/2, 2, 5/2). Of course, usually√ when two sides are rational the third side is not rational, such as the (1, 1, 2) right triangle. Any rational right triangle has a rational area, but not all (positive) rational numbers can occur as the area of a rational right triangle. For instance, no rational right triangle has area 1. This was proved by Fermat. The question we will examine here is: which rational numbers occur as the area of a rational right triangle? Deﬁnition 1.1. A positive rational number n is called a congruent number if there is a rational right triangle with area n: there are rational a, b, c > 0 such that a2 + b2 = c2 and (1/2)ab = n. This archaic use of the word congruent has nothing to do (directly) with congruences in modular arithmetic. The etymology will be explained in Section 3. The history of congruent numbers can be found in [2, Chap. XVI], where it is indicated that an Arab manuscript called the search for congruent numbers the “principal object of the theory of rational right triangles.” Example 1.2. The number 6 is congruent, being the area of the (3, 4, 5) right triangle. Example 1.3. The (9, 40, 41) right triangle has area 180 = 5·62, so dividing the lengths by 6 produces the rational right triangle (3/2, 20/3, 41/6) with area 5. That is, 5 is a congruent number. Example 1.4. The right triangle with sides (175, 288, 337) has area 25200 = 7·602, so scaling by 60 produces the right triangle (35/12, 24/5, 337/60) with area 7. Thus 7 is a congruent number. The congruent number problem asks for a description of all congruent numbers. Since scaling a triangle changes its area by a square factor, and every rational number can be multiplied by a suitable rational square to become a squarefree integer (e.g., 18/7 = 32 · 2/7, so multiplying by (7/3)2 produces the squarefree integer 14), we can focus our attention in the congruent number problem on squarefree positive integers. For instance, to 1 2 KEITH CONRAD say 1 is not a congruent number means no rational square is a congruent number. When n is squarefree in Z+, to show n is a congruent number we just need to ﬁnd an integral right triangle whose area has squarefree part n. Then writing the area as m2n shows scaling the sides by m produces a rational right triangle with area n.

2. A bad algorithm There is a parametric formula for primitive Pythagorean triples, and using it we will make a small list of squarefree congruent numbers. Any primitive triple (with even second leg) is (k2 − `2, 2k`, k2 + `2) where k > ` > 0, (k, `) = 1, and k 6≡ ` mod 2. In Table 1 we list such primitive triples where k + ` ≤ 9. The squarefree part of the area is listed in the last column. Each number in the fourth column is a congruent number and each number in the ﬁfth column is also a congruent number. The ﬁnal row of the table explains how Example 1.3 is found.

k ` (a, b, c) (1/2)ab Squarefree part 2 1 (3, 4, 5) 6 6 4 1 (15, 8, 17) 60 15 3 2 (5, 12, 13) 30 30 6 1 (35, 12, 37) 210 210 5 2 (21, 20, 29) 210 210 4 3 (7, 24, 25) 84 21 8 1 (63, 16, 65) 504 126 7 2 (42, 28, 53) 630 70 5 4 (9, 40, 41) 180 5 Table 1. Congruent Numbers

Notice 210 shows up twice in Table 1. Do other numbers which occur once also occur again? We will return to this question later. Table 1 can be extended according to increasing values of k + `, and any squarefree congruent number eventually shows up in the last column, e.g., the triangle (175, 288, 337) from Example 1.4, whose area has squarefree part 7, occurs at k = 16 and ` = 9. Alas, the table is not systematic in the appearance of the last column: we can’t tell by building the table when any particular number should occur, if at all, in the last column, so this method of generating (squarefree) congruent numbers is not a good algorithm. For instance, 53 is a congruent number, but it shows up for the ﬁrst time when k = 1873180325 and ` = 1158313156. (The corresponding right triangle has area 53 · 2978556542849787902.) Tabulations of congruent numbers can be found in Arab manuscripts from the 10th century, and 5 and 6 appear there. Fibonacci discovered in the 13th century that 7 is congruent and he stated that 1 is not congruent (that is, no THE CONGRUENT NUMBER PROBLEM 3 rational right triangle has area equal to a perfect square). The ﬁrst accepted proof is due to Fermat in the mid-1600s.

Theorem 2.1 (Fermat). The numbers 1 and 2 are not congruent.

Proof. In Table 2, the ﬁrst two columns show how to convert the sides (a, b, c) of a rational right triangle with area 1 into a positive rational solution of the equation y2 = x4 − 1 and conversely. (These correspondences are not inverses, but at least they show a positive rational solution in the ﬁrst column leads to a positive rational solution in the second column, and conversely.) The last two columns give a (bijective) correspondence between rational right triangles with area 2 and positive rational solutions of y2 = x4 + 1. So showing 1 and 2 are not congruent numbers is the same as showing the equations y2 = x4 ± 1 don’t have solutions in positive rational numbers.

a2 + b2 = c2, y2 = x4 − 1 a2 + b2 = c2, y2 = x4 + 1 1 1 2 ab = 1 2 ab = 2 x = c/2 a = y/x x = a/2 a = 2x y = |a2 − b2|/4 b = 2x/y y = ac/4 b = 2/x c = (x4 + 1)/xy c = 2y/x Table 2

A positive rational solution to y2 = x4 ± 1 can be turned into a positive integral solution of w2 = u4 ± v4 by clearing a common denominator. We can go in reverse by dividing by v4. That 1 and 2 are not congruent is therefore the same as the equations w2 = u4 ± v4 having no positive integer solutions. The reader is referred to [1, pp. 252–256] for a proof by descent that w2 = u4 ± v4 has no positive integer solutions. (Fermat discovered his method of descent on exactly this problem.)

Remark 2.2. Fermat considered the equations w2 = u4 ± v4 merely as auxiliary tools to get the proof of Theorem 2.1, but nowadays these equations are more prominent than Theorem 2.1 itself since the equations w2 = u4 ±v4 are related to the exponent 4 case of Fermat’s Last Theorem. √ √ Theorem 2.1√ leads√ to two weird proofs that 2 is irrational. If 2 were rational then 2, 2, and 2 are the sides of a rational right triangle with area√ 1. This is a contradiction of 1 not√ being a congruent number! Similarly, if 2 were rational then 2, 2, and 2 2 would be the sides of a rational right triangle with area 2, but 2 is not a congruent number! Fermat showed 3 is √not a congruent number, and the reader is invited to deduce from this that 3 is irrational. 4 KEITH CONRAD

3. Relation to Arithmetic Progressions of Three Squares The three squares 1, 25, 49 form an arithmetic progression with common difference 24. The squarefree part of 24 is 6. This is related to 6 being a congruent number, by the following theorem. Theorem 3.1. Let n > 0. There is a one-to-one correspondence between right triangles with area n and 3-term arithmetic progressions of squares with common difference n: the sets {(a, b, c) : 0 < a < b < c, a2 + b2 = c2, (1/2)ab = n} and {(r, s, t) : 0 < r < s < t, s2 − r2 = n, t2 − s2 = n} are in one-to-one correspondence by (a, b, c) 7→ ((b − a)/2, c/2, (b + a)/2), (r, s, t) 7→ (t − r, t + r, 2s). Proof. It is left to the reader to check the indicated functions take values in the indicated sets, and that the correspondences are inverses of one another: if you start with an (a, b, c) and make an (r, s, t) from it, and then form an (a0, b0, c0) from this (r, s, t), you get back the original (a, b, c). Similarly, starting with an (r, s, t), producing an (a, b, c) from it and then producing 0 0 0 an (r , s , t ) from that returns the same (r, s, t) you started with. When n > 0 is rational, the correspondence in Theorem 3.1 preserves rationality: (a, b, c) is a rational triple if and only if (r, s, t) is a rational triple. Therefore n is congruent if and only if there is a rational square s2 such that s2 − n and s2 + n are also squares. Note the correspondence in Theorem 3.1 involves not the squares in arithmetic progression but their positive rational square roots r, s, and t. Example 3.2. For n = 6, using (a, b, c) = (3, 4, 5) in Theorem 3.1 produces (r, s, t) = (1/2, 5/2, 7/2), whose termwise squares are the arithmetic progression 1/4, 25/4, 49/4 with common difference 6. Example 3.3. Taking n = 5 and (a, b, c) = (3/2, 20/3, 41/6), the correspondence in Theorem 3.1 yields (r, s, t) = (31/12, 41/12, 49/12): the rational squares (31/12)2, (41/12)2, (49/12)2 are an arithmetic progression with common difference 5. Example 3.4. Since 1 and 2 are not congruent numbers, there is no arithmetic progression of 3 rational squares with common difference 1 or 2 (or, more generally, common difference a nonzero square or twice a nonzero square). We now can explain the origin of the peculiar name “congruent number.” Fibonacci, in his book Liber Quadratorum (Book of Squares) from 1225, called an integer n a congruum if there is an integer x such that x2 ± n are both squares. This means x2−n, x2, x2+n is a 3-term arithmetic progression of squares. Fibonacci’s motivation for writing his book was the study of THE CONGRUENT NUMBER PROBLEM 5

3-term arithmetic progressions of integral (rather than rational) squares. Both words congruum and congruence come from the Latin congruere, which means “to meet together” (to congregate!). A congruum is a number related to three integer squares in a kind of agreement (having a common diﬀerence). Considering a congruum multiplied by rational squares (e.g., 24·(1/2)2 = 6) gives the congruent numbers.

4. The Curve y2 = x3 − n2x Whether or not n is congruent is related to solvability of pairs of equations: first, by definition we need to solve a2 + b2 = c2 and (1/2)ab = n in positive rational numbers a, b, and c. In Section 3 we saw this is equivalent to solving a second pair of equations in positive rational numbers: s2 − r2 = n and t2 − s2 = n. It turns out that the congruent number property is also equivalent to (nontrivial) rational solvability of the single equation y2 = x3 − n2x. This equation has three obvious rational solutions: (0, 0), (n, 0), and (−n, 0). These are the solutions with y = 0. Theorem 4.1. For n > 0, there is a one-to-one correspondence between the following two sets: {(a, b, c): a2 + b2 = c2, (1/2)ab = n}, {(x, y): y2 = x3 − n2x, y 6= 0}. Mutually inverse correspondences between these sets are nb 2n2 x2 − n2 2nx x2 + n2 (a, b, c) 7→ , , (x, y) 7→ , , . c − a c − a y y y Proof. This is a direct calculation left to the reader. We divide by c − a in the first formula, and c 6= a automatically since if c = a then b = 0, but (1/2)ab = n is nonzero. Restricting y to a nonzero value is necessary since we divide by y in the second formula. Remark 4.2. It is of course natural to wonder how the correspondence in Theorem 4.1 could be discovered in the first place. See the appendix. When n > 0 is rational, the correspondence in Theorem 4.1 preserves rationality: (a, b, c) is rational if and only if (x, y) is rational. Also, (a, b, c) is positive if and only if (x, y) is positive. (If a, b, and c are positive then (c − a)(c + a) = b2 > 0, so c − a is positive and thus x = nb/(c − a) > 0 and y = 2n2/(c − a) > 0. In the other direction, if x and y are positive then from y2 = x3 − n2x = x(x2 − n2) we see x2 − n2 has to be positive, so a, b, and c are all positive.) Any solution to a2 + b2 = c2 and (1/2)ab = n needs a and b to have the same sign (since ab = 2n > 0), and by a sign adjustment there is a rational solution with a, b, and c all positive if there is any rational solution at all. Therefore n is congruent if and only if the equation y2 = x3 − n2x has a rational solution (x, y) with y 6= 0; we don’t have to pay attention to whether or not x and y are positive. 6 KEITH CONRAD

A positive rational number n is not congruent if and only if the only rational solutions to y2 = x3−n2x have y = 0: (0, 0), (n, 0), and (−n, 0). For example, since 1 and 2 are not congruent (Theorem 2.1), the only rational solutions to y2 = x3 − x and to y2 = x3 − 4x have y = 0. Example 4.3. Since 6 is the area of a (3, 4, 5) right triangle, the equation y2 = x3 −36x has a rational solution with y 6= 0. The solution corresponding to the (3, 4, 5) right triangle by Theorem 4.1 is (x, y) = (12, 36). Example 4.4. From the rational right triangle (3/2, 20/3, 41/6) with area 5, Theorem 4.1 gives us a rational solution to y2 = x3 − 25x:(x, y) = (25/4, 75/8). If we allow sign changes on the coordinates of (3/2, 20/3, 41/6), Theorem 4.1 will give us new rational solutions to y2 = x3 − 25x. Using the triples of the form (±3/2, ±20/3, ±41/6) where the ﬁrst two coordinates have the same sign, the new solutions we get to y2 = x3 − 25x are collected in Table 3.

Signs on (3/2, 20/3, 41/6) (x, y) (+, +, +) (25/4, 75/8) (+, +, −) (−4, −6) (−, −, +) (−4, 6) (−, −, −) (25/4, −75/8) Table 3. Solutions to y2 = x3 − 25x

Example 4.5. A rational solution to y2 = x3−49x is (25, 120). Theorem 4.1 produces from this solution the rational right triangle (24/5, 35/12, 337/60) with area 7, which we met already in Example 1.4. Example 4.6. In Table 1 we found two rational right triangles with area 210: (35, 12, 37) and (21, 20, 29). Using Theorem 4.1, these triangles lead to two rational solutions to y2 = x3 − 2102x: (1260, 44100) and (525, 11025), respectively. An additional solution is (x, y) = (240, 1800), which leads to the rational right triangle (15/2, 56, 113/2) with area 210. We have seen that the following properties of a positive rational number n are equivalent: • there is a rational right triangle with area n, • there is a 3-term arithmetic progression of (nonzero) rational squares with common diﬀerence n, • there is a rational solution to y2 = x3 − n2x with y 6= 0. As an indication of what the viewpoint of the equation y2 = x3 − n2x is good for, its graph in the plane is a curve and we can use the geometry of this curve to do something striking: produce new rational right triangles with area n from old ones. (There is nothing in the deﬁnition of a congruent number which suggests this is even possible, although 210 shows up twice THE CONGRUENT NUMBER PROBLEM 7

2 3 2 in Table 1.) Given a rational point (x0, y0) on y = x − n x with y0 6= 0, draw the tangent line to this curve at the point (x0, y0). This line will meet the curve in a second rational point, and that can be converted into a new rational right triangle with area n using the correspondence of Theorem 4.1. This tangent process can be repeated. Example 4.7. Take n = 6. We will use tangent lines to points on the curve y2 = x3 − 36x to produce new rational points from old one, thereby getting new rational right triangles with area 6. The (3, 4, 5) right triangle with area 6 corresponds to the point (12, 36) on the curve y2 = x3 − 36x, as we saw already in Example 4.3. The tangent line to this curve at the point (12, 36) is y = (11/2)x − 30, which meets the curve in the second point (25/4, 35/8). The tangent line to the curve at (25/4, 35/8) has equation 1299 6005 y = x − , 140 112 which meets the curve in the new point 1442401 1726556399 (4.1) , . 19600 2744000 Using Theorem 4.1, (25/4, 35/8) corresponds to the rational right triangle of area 6 having sides (7/10, 120/7, 1201/70). The rational right triangle with area 6 corresponding to (4.1) has sides 1437599 2017680 2094350404801 (4.2) , , . 168140 1437599 241717895860 Armed with 3 rational right triangles with area 6, we can find 3 arithmetic progressions of rational squares using Theorem 3.1. The (3, 4, 5) triangle, as we saw in Example 3.2, yields the arithmetic progression 1/4, 25/4, 49/4. The (7/10, 120/7, 1201/70) right triangle yields the arithmetic progression 11512 12012 12492 , , . 140 140 140 The right triangle with sides in (4.2) yields the arithmetic progression 17274381696012 20943504048012 77611083871 2 , , . 483435791720 483435791720 483435791720 All of these arithmetic progressions of squares have common difference 6. This tangent method of finding new rational points on y2 = x3 −n2x from old ones can be extended to a secant method: the line through two rational points on the curve will meet the curve in a third rational point. Example 4.8. Suppose (a, b, c) satisfies a2 +b2 = c2 and (1/2)ab = n. Such a solution gives rise to seven additional ones: (−a, −b, −c) and (a, b, −c), (−a, −b, c), (b, a, c), (b, a, −c), (−b, −a, c), (−b, −a, −c). 8 KEITH CONRAD

These algebraic modiﬁcations have a geometric interpretation in terms of constructing new points from old ones on the curve y2 = x3 − n2x with y 6= 0. Say (a, b, c) corresponds to (x, y) by Theorem 4.1. From the point (x, y) on the curve we get automatically a second point: (x, −y). This corresponds by Theorem 4.1 to (−a, −b, −c). What points on the curve correspond to the six remaining algebraic modiﬁcations above? Well, there are three obvious points on the curve which have nothing to do with our particular (x, y), namely (0, 0), (n, 0), and (−n, 0). The line through (x, y) and (0, 0) meets the curve in the point (−n2/x, −n2y/x2), which corresponds by Theorem 4.1 to (a, b, −c). More generally, the three lines through (x, y) and each of (0, 0), (n, 0), and (−n, 0) meet the curve in three additional points. Similarly, the three lines through (x, −y) and each of (0, 0), (n, 0), and (−n, 0) meet the curve in three additional points. See Table 4. The corresponding triples from Theorem 4.1 are collected in Table 5 and are exactly what we were looking for.

First Point Second Point Third Point (x, y) (0, 0) (−n2/x, −n2y/x2) (x, −y) (0, 0) (−n2/x, n2y/x2) (x, y) (n, 0) (n(x + n)/(x − n), 2n2y/(x − n)2) (x, −y) (n, 0) (n(x + n)/(x − n), −2n2y/(x − n)2) (x, y) (−n, 0) (−n(x − n)/(x + n), 2n2y/(x + n)2) (x, −y) (−n, 0) (−n(x − n)/(x + n), −2n2y/(x + n)2) Table 4. Third Intersection Point of a Line with y2 = x3 − n2x

Pair Triple (x, y) (a, b, c) (x, −y) (−a, −b, −c) (−n2/x, −n2y/x2) (a, b, −c) (−n2/x, n2y/x2) (−a, −b, c) (n(x + n)/(x − n), 2n2y/(x − n)2) (b, a, c) (n(x + n)/(x − n), −2n2y/(x − n)2) (−b, −a, −c) (−n(x − n)/(x + n), 2n2y/(x + n)2) (−b, −a, c) (−n(x − n)/(x + n), −2n2y/(x + n)2) (b, a, −c) Table 5. Theorem 4.1 and Sign Changes

Remark 4.9. The secant method is a way to “add” points and the tangent method is essentially the special case of “doubling” a point. These tangent and secant constructions can be used to give the rational points on y2 = x3 − n2x the structure of an abelian group in which, for rational n > 0, any rational point (x, y) with y 6= 0 has infinite order. (This is not at all THE CONGRUENT NUMBER PROBLEM 9 obvious.) Therefore the curve y2 = x3 − n2x has infinitely many rational points as soon as it has just one rational point with y 6= 0, so there are infinitely many right triangles with area n provided there is one example and there are infinitely many 3-term arithmetic progressions of rational squares with common difference n provided there is one example. In terms of Table 1, any area arising in the table at least once will arise in the table infinitely often.1 The importance of thinking about congruent numbers in terms of the curves y2 = x3 − n2x goes far beyond this interesting construction of new rational right triangles with area n from old ones: this viewpoint in fact leads to a tentative solution of the whole congruent number problem! In 1983, Tunnell [7] used arithmetic properties of y2 = x3 − n2x to discover a previously unknown elementary necessary condition on congruent numbers and he was able to prove the condition is sufficient if a certain other conjecture is true. Theorem 4.10 (Tunnell). Let n be a squarefree positive integer. Set f(n) = #{(x, y, z) ∈ Z3 : x2 + 2y2 + 8z2 = n}, g(n) = #{(x, y, z) ∈ Z3 : x2 + 2y2 + 32z2 = n}, h(n) = #{(x, y, z) ∈ Z3 : x2 + 4y2 + 8z2 = n/2}, k(n) = #{(x, y, z) ∈ Z3 : x2 + 4y2 + 32z2 = n/2}. For odd n, if n is congruent then f(n) = 2g(n). For even n, if n is congruent then h(n) = 2k(n). Moreover, if the weak Birch and Swinnerton–Dyer conjecture is true for the curve y2 = x3 − n2x then the converse of both implications is true: f(n) = 2g(n) implies n is congruent when n is odd and h(n) = 2k(n) implies n is congruent when n is even. The weak Birch and Swinnerton–Dyer conjecture, which we won’t describe here, is one of the most important conjectures in mathematics. (It is on the list of seven Clay Millenium Prize problems.) Several years be- fore Tunnell proved his theorem, Stephens [6] showed the weak Birch and Swinnerton–Dyer conjecture implies any squarefree positive integer n ≡ 5, 6, 7 mod 8 is a congruent number.2 Tunnell’s achievement was discovering the enumerative criterion for congruent numbers and its relation to the weak Birch and Swinnerton–Dyer conjecture. For background on the ideas in Tunnell’s theorem, see [3] and [4]. In [5, pp. 112–114] the particular case of prime congruent numbers is considered. Tunnell’s theorem provides an unconditional method of proving a squarefree positive integer n is not congruent (show f(n) 6= 2g(n) or h(n) 6= 2k(n), depending on the parity of n), and a conditional method of proving n is

1The two rational points on y2 = x3 − 2102x which correspond to the repetition of 210 in Table 1 are independent in the group law: they do not arise as multiples of some common rational point, nor do they have a common multiple. 2Inﬁnitely many nonsquarefree positive n ≡ 5, 6, 7 mod 8 are not congruent numbers. 10 KEITH CONRAD congruent (conditional, that is, on the weak Birch and Swinnerton-Dyer conjecture for the curve y2 = x3 − n2x). Example 4.11. Since f(1) = g(1) = 2 and f(3) = g(3) = 4, we have f(n) 6= 2g(n) for n = 1 and 3, so Tunnell’s criterion shows 1 and 3 are not congruent. Example 4.12. Since h(2) = k(2) = 2, we have h(2) 6= 2k(2), so Tunnell’s criterion shows 2 is not congruent. Example 4.13. Since f(5) = g(5) = 0 and f(7) = g(7) = 0, we have f(n) = 2g(n) for n = 5 and 7. Tunnell’s theorem says 5 and 7 are congruent if the weak Birch and Swinnerton-Dyer conjecture is true for y2 = x3 − 25x and y2 = x3 − 49x. Unconditionally, we saw earlier that 5 and 7 are congruent. Example 4.14. Since h(10) = 4 and k(10) = 4, h(10) 6= 2k(10), so Tun- nell’s theorem says 10 is not a congruent number. Example 4.15. We will show (conditionally) that any squarefree positive integer n satisfying n ≡ 5, 6, 7 mod 8 is a congruent number. Tunnell’s theorem tells us to check that f(n) = 2g(n) when n ≡ 5, 7 mod 8 and h(n) = 2k(n) when n ≡ 6 mod 8. Since x2 + 2y2 6≡ 5, 7 mod 8 for any integers x and y, f(n) = 0 and g(n) = 0 when n ≡ 5, 7 mod 8, so f(n) = 2g(n). When n ≡ 6 mod 8 we have n/2 ≡ 3 mod 4, so x2 6≡ n/2 mod 4 for any integer x. Therefore h(n) = 0 and k(n) = 0 when n ≡ 6 mod 8, so h(n) = 2k(n). This shows n is congruent if the weak Birch and Swinnerton-Dyer conjecture is true for y2 = x3 − n2x.

Appendix A. Discovering Theorem 4.1 Fix a number n 6= 0. The real solutions (a, b, c) to each of the equations 1 (A.1) a2 + b2 = c2, ab = n, 2 describe a surface in R3, so it is reasonable to expect these two surfaces intersect in a curve. We want an equation for that curve, which will be y2 = x3 − n2x in the right choice of coordinates. Two approaches will be described, one algebraic and the other geometric. The sign on n will be irrelevant, so we allow any n 6= 0 rather than n > 0. The algebra is simpliﬁed by introducing a cross-term in the equation a2 + b2 = c2. Let c = t + a, which turns this equation into b2 = t2 + 2at, or equivalently (A.2) 2at = b2 − t2. Since ab = 2n is nonzero, neither a nor b is 0, so we can write a = 2n/b and substitute it into (A.2): 4nt = b2 − t2. b THE CONGRUENT NUMBER PROBLEM 11

Multiplying through by b makes this 4nt = b3 − t2b. Divide by t3 (t 6= 0, as otherwise a = c and then b = 0, but ab = 2n 6= 0):

4n b3 b = − . t2 t t Multiply through by n3:

2n2 2 nb3 nb = − n2 . t t t Set x = nb/t and y = 2n2/t, so y2 = x3 − n2x. Then x = nb/(c − a) and y = 2n2/(c − a), as in Theorem 4.1. We now turn to a geometric explanation of Theorem 4.1, taking greater advantage of the interpretation of the two equations in (A.1) as surfaces which meet in a curve. Rather than working with the equations as surfaces in R3, we will work in the projective space P3(R) by homogenizing the two equations. This doesn’t change the first equation in (A.1), but makes the second one (1/2)ab = nd2. Letting [a, b, c, d] be the homogeneous coordinates of a typical point in P3(R), the two equations 1 (A.3) a2 + b2 = c2, ab = nd2 2 each define surfaces in P3(R). Let C be the intersection of these surfaces (a curve). There are points on C with b = 0, namely [a, b, c, d] = [1, 0, ±1, 0]. These points are not in the usual affine space inside P3(R), and we will use one of these points in a geometric construction. Let’s project through the point P := [1, 0, 1, 0] to map C to the plane Π := {[0, b, c, d]} and find the equation for the image of C in this plane. The point P lies on C and not in Π. For each Q ∈ C other than P , the line PQ in P3(R) meets Π in a unique point. Call this point f(Q). When Q = P , intersect the tangent line to C at P with the plane Π to define f(P ). We have defined a function f : C → Π. Computing a formula for f necessitates a certain about of computation to see what happens. Suppose first that Q = [a, b, c, d] is not P . The line through P and Q is the set of points [λ + µa, µb, λ + µc, µd], which meets Π where λ = −µa, making f(Q) = [0, µb, µ(c − a), µd] = [0, b, c − a, d]. 12 KEITH CONRAD

As for f(P ), the tangent planes to each of the surfaces a2 + b2 = c2 and (1/2)ab = nd2 in P3(R) at the point P are the planes a = c and b = 0, so the tangent line at P is the set of points [a, 0, a, d], which meets Π in [0, 0, 0, 1], so f(P ) = [0, 0, 0, 1]. Thus ( [0, b, c − a, d], if [a, b, c, d] 6= [1, 0, 1, 0], f([a, b, c, d]) = [0, 0, 0, 1], if [a, b, c, d] = [1, 0, 1, 0]. As an exercise, check f is injective. (Hint: Since (1/2)ab = nd2, knowing b and d determines a if b 6= 0.) All points in the plane Π have first coordinate 0. Identify Π with P2(R) by dropping this coordinate, which turns f into the function g : C → P2(R) where ( [b, a − c, d], if [a, b, c, d] 6= [1, 0, 1, 0], (A.4) g([a, b, c, d]) = [0, 0, 1], if [a, b, c, d] = [1, 0, 1, 0]. We have mapped our curve C to the projective plane P2(R). What is an equation for the image g(C)? For Q = [a, b, c, d] on C, write g(Q) = [x, z, y]. (This arrangement of letters will make formulas come out in close to the expected way more quickly.) When Q 6= [1, 0, 1, 0] (that is, a 6= c), (A.4) says x = b, y = d, and z = c − a 6= 0.3 The equations in (A.3) become a2 + x2 = (a + z)2 and (1/2)ax = ny2, so x2 = 2az + z2, ax = 2ny2. Since z 6= 0, we can solve for a in the first equation, so a is determined by x, y, and z. Multiplying the first equation by x and the second by 2z, x3 = 2axz + xz2 = 4ny2z + xz2. Thus 4ny2z = x3 − xz2. Set X = x, Y = 2ny, and Z = z/n to find Y 2Z = X3 − n2XZ2, which is the homogeneous form of Y 2 = X3 − n2X. Tracing this correspondence out explicitly from the start, if we begin with [a, b, c, d] on C where d 6= 0 (the standard affine part of C), its image [X,Z,Y ] in P2(R) is c − a nb 2n2d b, , 2nd = [nb, c − a, 2n2d] = , 1, . n c − a c − a Since d 6= 0 implies a 6= c, in inhomogeneous coordinates with middle coordinate d = 1 the point (a, b, c) goes to (nb/(c − a), 2n2/(c − a)), which is the transformation in Theorem 4.1. As an exercise in these techniques, consider the problem of classifying triangles with a given area n > 0 and a given angle θ. (Taking θ = π/2

3The cross term t = c − a in the algebraic method is precisely z, so now we get a geometric interpretation of this cross term as a coordinate in a projection map to a plane. THE CONGRUENT NUMBER PROBLEM 13 is the congruent number problem.) Let a, b, c be the side lengths of the triangle, with c the length of the edge opposite the angle θ. The equations in (A.1) are replaced by 1 (A.5) a2 + b2 − 2ab cos θ = c2, ab sin θ = n. 2 (If there is a solution with rational a, b, c, and n then cos θ and sin θ must be rational.) Show the solutions (a, b, c) of (A.5) are in one-to-one correspondence with the solutions (x, y) of the equation 2n cos θ y2 = x3 + x2 − n2x sin θ cos θ + 1 cos θ − 1 = x x + n x + n , sin θ sin θ with y 6= 0. The correspondence should specialize to that in Theorem 4.1 when θ = π/2.

Appendix B. Other Diophantine Equations In Table 2 we saw the congruent number property for 1 and 2 is equivalent to the solvability of a single equation in positive rational numbers, either y2 = x4 − 1 (for 1) or y2 = x4 + 1 (for 2). These each generalize: n is congruent if and only if y2 = x4 − n2 has a positive rational solution and if and only if y2 = x4 + 4n2 has a positive rational solution. (In particular, y2 = x4 − n2 has a positive rational solution if and only if y2 = x4 + 4n2 does.) See Table 6, where the ﬁrst two columns turn right triangles with area n into positive rational solutions of y2 = x4 −n2 and conversely, and the last two columns do the same with y2 = x4 + 4n2. The correspondences in the ﬁrst two columns are not inverses of each other, but the correspondences in the last two columns are inverses. (When n = 2 the equation in Table 6 is y2 = x4 + 16 rather than y2 = x4 + 1 as in Table 2. We can easily pass from the former to the latter by replacing y with 4y and x with 2x.) The equivalence of n being congruent with y2 = x4 − n2 having a positive rational solution is due to Lucas (1877).

a2 + b2 = c2, y2 = x4 − n2 a2 + b2 = c2, y2 = x4 + 4n2 1 1 2 ab = n 2 ab = n x = c/2 a = y/x x = a a = x y = |a2 − b2|/4 b = 2nx/y y = ac b = 2n/x c = (x4 + n2)/xy c = y/x Table 6

How would the equations y2 = x4−n2 or y2 = x4+4n2 be discovered? The arithmetic progression viewpoint leads to one of them. Rewriting s2 −r2 = n and t2 − s2 = n as r2 = s2 − n and t2 = s2 + n, multiplication gives (rt)2 = s4 − n2, so we’ve solved y2 = x4 − n2 in positive rational numbers. 14 KEITH CONRAD

Unlike solutions to y2 = x3−n2x, we are not claiming that the realizations of n as the area of a right triangle are in a one-to-one correspondence with all the positive rational solutions to y2 = x4 − n2 or y2 = x4 + 4n2. There is at least one rational right triangle with area n if and only if these equations have at least one positive rational solution. Remark B.1. For t 6= 0, solutions to y2 = x4 + t correspond to solutions to Y 2 = X3 − 4tX by (x, y) 7→ (2t/(y − x2), 4tx/(y − x2)) and (X,Y ) 7→ (Y/2X, (Y 2+8tX)/4X2). In particular, solutions to y2 = x4−n2 correspond to Y 2 = X3 +(2n)2X, which is not the equation Y 2 = X3 −(2n)2X and thus isn’t related to whether or not 2n is a congruent number. Explicit examples show the lack of a general connection between n and 2n being congruent: 5 is congruent but 10 is not, while 3 is not congruent but 6 is.

References [1] D. M. Burton, “Elementary Number Theory,” 6th ed., McGraw-Hill, New York, 2007. [2] L. E. Dickson, “History of the Theory of Numbers,” Vol. II, Chelsea, New York, 1952. [3] G. Henniart, Congruent Numbers, Elliptic Curves, and Modular Forms, translation by F. Lemmermeyer at http://www.fen.bilkent.edu.tr/∼franz/publ.html. [4] N. Koblitz, “Introduction to Elliptic Curves and Modular Forms,” 2nd ed., Springer– Verlag, New York, 1993. [5] A. Knapp, “Elliptic Curves,” Princeton Univ. Press, Princeton, 1992. [6] N. M. Stephens, Congruence properties of congruent numbers, Bull. London Math. Soc. 7 (1975), 182–184. [7] J. Tunnell, A Classical Diophantine Problem and Modular Forms of Weight 3/2, In- vent. Math. 72 (1983), 323–334.